Conditional Probability using the Law of Total Probability

From families with three children, a child is selected at random and found to be a girl. What is the probability that she has an older sister? Assume that in a three child family all sex distributions are equally probable.

The book then gives this hint: Let G be the event that the randomly selected child is a girl, A be the even that she has an older sister, and O, M, and Y be the events that she is the oldest, the middle, and the youngest child respectively. For any subset B of the sample space let ; then apply the Law of Total Probability to Q.

So I have the given information in the hint to start with and the possible combinations of three children. {bbb, bbg, bgb, gbb, bgg, gbg, ggb, ggg}.

The Law of Total Probability states:

I'm having a lot of trouble figuring out how to set this up and how to get started. I'm even questioning if I know what P(G) is for sure.

From families with three children, a child is selected at random and found to be a girl. What is the probability that she has an older sister? Assume that in a three child family all sex distributions are equally probable.

The book then gives this hint: Let G be the event that the randomly selected child is a girl, A be the even that she has an older sister, and O, M, and Y be the events that she is the oldest, the middle, and the youngest child respectively. For any subset B of the sample space let ; then apply the Law of Total Probability to Q.

So I have the given information in the hint to start with and the possible combinations of three children. {bbb, bbg, bgb, gbb, bgg, gbg, ggb, ggg}.

The Law of Total Probability states:

I'm having a lot of trouble figuring out how to set this up and how to get started. I'm even questioning if I know what P(G) is for sure.